E. Z3 Z4 - 1 - Journal de Physique Lettres

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et de sortie, corrigées par la perte de masse Qgg de la réaction étudiée, joue un ... Vi(R) de la voie d'entrée est située au-dessus de celui de la voie de sortie.






Laboratoire de Chimie Nucléaire, Institut de Physique Nucléaire, B.P. 1, 91406



Résumé. La possibilité de transferts sous-coulombiens est discutée du point de vue de l’énergétique de la réaction. La position relative des courbes d’énergie potentielle pour les voies d’entrée et de sortie, corrigées par la perte de masse Qgg de la réaction étudiée, joue un rôle déterminant. Lorsque le potentiel Vi(R) de la voie d’entrée est située au-dessus de celui de la voie de sortie Vf(R) - Qgg, le transfert est énergétiquement possible; il intervient dès que les noyaux sont suffisamment proches l’un de l’autre et cela même pour des énergies inférieures à la barrière. Une comparaison est faite avec des résultats expérimentaux. 2014

Abstract. The possibility of sub-barrier transfers is discussed from the point of view of the energetics of the reaction. The position of the entrance channel potential energy curve Vi(R) relative to that of the exit channel corrected for the mass balance Vf(R) - Qgg appears to be crucial. When the former curve is situated above the latter one, the transfer is energetically probable and will occur as soon as the colliding nuclei are sufficiently close to each other, i.e. for sub-barrier incident energies. Experimental examples are given. 2014

Considerable experimental and theoretical attention has recently been centered on questions that relate to thresholds of nuclear reactions and to the interaction barrier [1-5]. However, the notion of interaction barrier is still not quite clear. Generally, the interaction barrier is defined as the value Bi at which the interaction potential (nucleon + Coulomb) for the s-wave passes through a maximum; but this must be regarded as a strong interaction barrier, or a fusion barrier and some nuclear reactions, namely transfer reactions can take place at incident energies somewhat lower than Bi. The above definition, however, will be adopted here, and in considering the possibility of sub-barrier reactions, we shall ask the following question : for a given projectile target system, can one predict whether or not a given transfer reaction will occur with significant cross section for incident energies lower than Bi ? This question will be studied here from the point of view of the energetics involved in the reaction, which have been shown, in previous studies [6-13], to play a fundamental role. Many authors have underlined the influence of the Q value on reaction probabilities,


(*) Laboratoire de Chimie Nucleaire, Centre d’Etudes Nucleaires Bordeaux, 33170 Gradignan, France.

and the enhancements of transitions for which Q Qopt, where =

e~ Qopt


E.1 Z3 2Z4 - 11


1 (1)

c.m. incident energy, Zl Z2 the atomic numbers of the colliding nuclei and Z3 Z4 those of the transfer products [6-9]. Others [10-11] have discussed the dramatic influence on the transfer cross sections of Qgg - AP~, Qgg being the Q’value for a ground state transition, and d Y~ the difference between the Coulomb energies in the final and initial

Ei being the

states A~


Ve ~,.

Similar ideas are involved in the present paper, but firstly we shall focus on the feasibility of subbarrier reactions, and secondly we shall include a nuclear part in addition to Pc in the expression for the interaction potential. This nuclear contribution may have a non negligible influence for incident energies slightly lower than the interaction barrier. consider the + 209Bi. In figures 1 and 2 are shown the experimental excitation functions obtained by activation techniques [14, 15] for 21 opo and 211 At which correspond respectively to transfers of 1 and 2 protons from the projectile 1.



systems 14N

evidence. +




209Bi and 160

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyslet:0197500360305900


to the

target. The excitation function for 211Rn

be seen in figures 1 and 2 that the relative of single-proton and two-proton transfer excitation functions for the two systems are quite different. With a nitrogen beam, the lp-transfer reaction has a significant cross section for energies below the interaction barrier. This is not the case for the 2p-transfer cross section. On the contrary, with the oxygen beam, both excitation functions rise somewhat below the interaction barrier. It

is also plotted in figure 1. From angular distributions and recoil range measurements, it was established in reference [14] that this production is made through the compound nucleus reaction : 14N + 209Bi --+ 2 19Th + 4n followed by two fast a decays. Therefore, in order to induce this reaction the projectile must overcome the strong interaction barrier. The vertical solid line indicates the value of the interaction barrier calculated with a WoodsSaxon nuclear potential (see below).






FIG. 2. Excitation functions for the reactions 16O + 209Bi~210pO (circles) and 211At (squares). Open symbols are data from reference [14], black symbols are from reference [15]. The lines are drawn to guide the eye. The vertical line indicates the interaction -


In order to explain 2. Potential energy curves. must consider the energetics such behaviour, one the incident c.m. energy Ei reactions : of the particular can be divided into a potential energy Vi(R) and a kinetic energy Eki(R), according to the distance R between the centers of the nuclei. The interaction potential I~;(R) is the sum of a repulsive Coulomb part Vci(R) and an attractive nuclear part VNi(R). Only head-on collisions (I 0) will be considered in this paper which deals with reactions induced ~ or below the barrier. The centrifugal part Vce(l, R ) near of the interaction potential will therefore be ignored. -

Excitation functions for the reactions 14N + 209Bi~210po ( ~ ) and 211 Rn (~), corresponding respectively to lp-transfer, 2p-transfer, and a compound nucleus process. The lines are drawn to guide the eye. The vertical line indicates the interaction barrier. The data are from reference [14]. FIG. 1.

(~ ),






The general shape of figure 3. Its maximum action barrier Bi.

the curve Vi(R) is shown in defines the value of the inter-

The existence of a crossing point for the curves (i) and ( f - Q) can be discussed in terms of the values of Qgg and MJ = Bf - Bi (see Fig. 3). ~ If Bi > Bf -Qgg, i.e.



crossing point will exist at a distance of approach Rcp which is larger than the distance corresponding to the interaction barrier, RB. For R Rcp, condition (3) will be satisfied. Therefore, the energy Ecp ~,(J~p) can be considered as the energetic threshold for the transfer reaction and for incident energies higher than Ecp, the transfer will be energetically favoured. However, the reaction will be effectively observed only if the proximity condition (a) is also fulfilled. Therefore, 2~p must be taken as a lower limit for the reaction threshold (see the restrictions of remark 1 below). If the value of J5cp is much lower than B, the reaction will be produced as soon as condition (a) FIG. 3. Interaction potentials in the entrance and exit channels is satisfied, and significant sub-barrier cross sections (see text). will be observed. ~ If Qgg - dB 0, no crossing point exists for If Ei Bi, the two nuclei will not be able to come R > RB and the transfer reaction will not have a closer than a certain distance ~ at which Eki 0. significant cross section below the barrier. The transfer reaction will occur with a significant Two remarks can be made about the previous probability if two conditions are fulfilled : a





a) Proximity condition : the nuclei must be close enough to each other to have the possibility of exchanging matter. b) Energy condition : the reaction must be energetically probable. As will be shown in what follows, this second condition is directly related to the relative position of the interaction potentials in the entrance and exit channels. For the purpose of simplicity, it is assumed that the transfer occurs only at the closest distance of approach R ;, the nuclei being at rest in the c.m. system (Eki 0). The distance Rf between the centers of the nuclei after transfer is assumed to be equal to R;, and the kinetic energy Ekf equal to zero. These assumptions neglect recoil and quantum effects. The energy balance can then be written as =

Vi(R.¡) + Qgg = vf(R~) where V;






the interaction potentials in the channels, and Ef the excitation


entrance and exit

energy of the final nuclei. The reaction will be Ff" ~ 0. This condition is

energetically possible equivalent to

v~ 1> Vf - 6gg


Remark 1. Because of the assumptions neglecting recoil and quantum effects, the value Ecp defined above must not be taken as an effective threshold under which the reaction cannot occur : the reaction can be induced for energies slightly lower than Ecp, if for example Eki =1= 0 or if Re =1= Ri. But the incident energy Ei Ecp should correspond to a sharp increase in the cross section. -


If one neglects the nuclear part of Remark 2. the interaction potential, the quantity Qgg - AB becomes Qgg - AEc, and one can understand the relevance of this quantity as far as transfer probabilities are concerned [ 10-11 ] . Moreover, condition (3) then becomes -

I This condition will be fulfilled tance of

minimum dis,


such that

Ei ~i

Z3 Z4 , e2 ~~ " gg Qgg ’. 201320132013~201320132013




and the solution of this inequality can be obtained by comparing the curves V;(R), (denoted by (i) in figure 3) and Vf(R) Qgg denoted by ( f - Q). For example, if charge is transferred from projectile to target, the curve Vf(R) is always situated below the curve Vi(R). If Qgg is negative, the relative positions of the relevant curves are as given in figure 3.

at the

approach R; for incident energies F, Zl Z2 e 2IRi

This inequality is eq. ( 1 )).



Qopt 6gg (see

Therefore, an approximate value E:p of the energy corresponding to the crossing point can be found by solving the following equation :.:


Refinements can be made to take into account recoil effects in the expression for Qopt.




favoured 3.


figure 4 that the transfer of one of 14N on 209Bi is energetically for very low incident energies


proton in the




The potential energy curves have been drawn for the transfer reactions mentioned in section 1. The following expressions were used for the entrance channel to

experimental data.

v~~(R cIR )


e2 = Z1 Z2 R

where Zi and Z2 denotes atomic jectile and target and

numbers of pro-

~ _



d VNI(R)



I + exp

(Woods-Saxon form) with Vo (depth parameter) 0.6 fm, 50 MeV, d (diffuseness parameter) and RT and + A1 A2 being respecA2 ~3) ro(A 1 1/3 tively the atomic masses of projectile and target, =




and ro Such


1.22 fm.

expression for the interaction potential gives a correct description of the outer surface of the nucleus which is of interest here [16]. The values adopted for the nuclear parameters are not critical, an

and the conclusions which follow by small changes in these values.


not affected

FIG. 5.




4 but for transfer reactions induced with 160 on 209Bi.


The only limitation to proton transfer is therefore the proximity condition. This case thus involves high cross sections below the barrier, as can be seen in figure 1. In contrast, the energetic conditions are much more critical for the 2p-transfer reaction : (Ecp 65 MeV, Qgg - AB 1.2 MeV) ; the crossing occurs only close to the barrier Bi 68 MeV. This restriction is well reflected in experimental data =



(Fig. 1).

Potential energy curves involved in the transfers of one and two protons induced with 14N on 209Bi. The interaction potential Vi is plotted versus the distance of approach for the entrance channel, and the quantities Vf - Qgg are plotted for the exit channels. FIG. 4.


As far as the exit channel is concerned, the quantity Vf(R) - Qgg was calculated with similar expressions for Vcf(R) and VNf(R) to those given above.

For the projectile 160, the behaviours of the potential energy curves corresponding to one- and twoproton transfers are very similar to each other (see Fig. 1). The barrier Bi is at 77 MeV, the crossings are observed at 2~p ~ 64 MeV for both reactions, and the values of Qgg - 0.8 are respectively 2 MeV (lp-transfer) and 3.6 MeV (2p-transfer). This is in agreement with the similarities observed in the shapes and positions of the experimental excitation functions

(Fig. 2). In conclusion, we note that from the data discussed here, the comparison of potential energy curves appears to be a good guide in predicting the feasibility

of sub-barrier transfer reactions. We are very grateful to M. Lefort for helpful discussions, and to F. Plasil for a careful reading of the manuscript.



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[15] CROFT,